Give a Venn-Diagram for each of the following assume A, B, C ⊆ U: (Ac ∪ Bc ∪ Cc) ∩ ((A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C))

Steps to Solve

1. Identify the Sets and Their Complements

Define the sets A, B, and C within the universal set U. Then, determine the complements of these sets:

• The complement of A ($A'$) includes all elements in U that are not in A.
• The complement of B ($B'$) includes all elements in U that are not in B.
• The complement of C ($C'$) includes all elements in U that are not in C.

2. Determine the Intersections

Calculate the intersections of the sets. The intersections needed are:

• $A \cap B$: Elements common to both A and B.
• $A \cap C$: Elements common to both A and C.
• $B \cap C$: Elements common to both B and C.
• $A \cap B \cap C$: Elements common to A, B, and C.

3. Visualize Using a Venn Diagram

Create a Venn Diagram that shows all three sets A, B, and C. In the diagram:

• Shade or highlight the regions corresponding to the intersections and their complements as needed to illustrate the specific scenarios.
• Ensure the universal set U is represented around the three circles to show all possible elements.

4. Labeling the Venn Diagram

Clearly label each set and intersection in the Venn Diagram. Include complementary areas. For instance:

• Label sections of the diagram with $A$, $B$, and $C$ alongside their complements.
• Indicate the intersections such as $A \cap B$, $A \cap C$, and $B \cap C$ in the respective areas.

5. Final Check and Interpretation

Review the Venn Diagram to ensure all intersections and their complements are accurately represented. Make sure the relationships between the sets are clear, and any specific elements required by the question are correctly highlighted.